341,821 research outputs found
Online Multivariate Changepoint Detection: Leveraging Links With Computational Geometry
The increasing volume of data streams poses significant computational
challenges for detecting changepoints online. Likelihood-based methods are
effective, but their straightforward implementation becomes impractical online.
We develop two online algorithms that exactly calculate the likelihood ratio
test for a single changepoint in p-dimensional data streams by leveraging
fascinating connections with computational geometry. Our first algorithm is
straightforward and empirically quasi-linear. The second is more complex but
provably quasi-linear: for data points.
Through simulations, we illustrate, that they are fast and allow us to process
millions of points within a matter of minutes up to .Comment: 31 pages,15 figure
Convex Hulls of Curves: Volumes and Signatures
Taking the convex hull of a curve is a natural construction in computational
geometry. On the other hand, path signatures, central in stochastic analysis,
capture geometric properties of curves, although their exact interpretation for
levels larger than two is not well understood. In this paper, we study the use
of path signatures to compute the volume of the convex hull of a curve. We
present sufficient conditions for a curve so that the volume of its convex hull
can be computed by such formulae. The canonical example is the classical moment
curve, and our class of curves, which we call cyclic, includes other known
classes such as -order curves and curves with totally positive torsion. We
also conjecture a necessary and sufficient condition on curves for the
signature volume formula to hold. Finally, we give a concrete geometric
interpretation of the volume formula in terms of lengths and signed areas.Comment: 15 pages, 5 figures. Comments are welcome
A Moving Boundary Flux Stabilization Method for Cartesian Cut-Cell Grids using Directional Operator Splitting
An explicit moving boundary method for the numerical solution of
time-dependent hyperbolic conservation laws on grids produced by the
intersection of complex geometries with a regular Cartesian grid is presented.
As it employs directional operator splitting, implementation of the scheme is
rather straightforward. Extending the method for static walls from Klein et
al., Phil. Trans. Roy. Soc., A367, no. 1907, 4559-4575 (2009), the scheme
calculates fluxes needed for a conservative update of the near-wall cut-cells
as linear combinations of standard fluxes from a one-dimensional extended
stencil. Here the standard fluxes are those obtained without regard to the
small sub-cell problem, and the linear combination weights involve detailed
information regarding the cut-cell geometry. This linear combination of
standard fluxes stabilizes the updates such that the time-step yielding
marginal stability for arbitrarily small cut-cells is of the same order as that
for regular cells. Moreover, it renders the approach compatible with a wide
range of existing numerical flux-approximation methods. The scheme is extended
here to time dependent rigid boundaries by reformulating the linear combination
weights of the stabilizing flux stencil to account for the time dependence of
cut-cell volume and interface area fractions. The two-dimensional tests
discussed include advection in a channel oriented at an oblique angle to the
Cartesian computational mesh, cylinders with circular and triangular
cross-section passing through a stationary shock wave, a piston moving through
an open-ended shock tube, and the flow around an oscillating NACA 0012 aerofoil
profile.Comment: 30 pages, 27 figures, 3 table
Mechanistic and pathological study of the genesis, growth, and rupture of abdominal aortic aneurysms
Postprint (published version
On pore-scale modeling and simulation of reactive transport in 3D geometries
Pore-scale modeling and simulation of reactive flow in porous media has a
range of diverse applications, and poses a number of research challenges. It is
known that the morphology of a porous medium has significant influence on the
local flow rate, which can have a substantial impact on the rate of chemical
reactions. While there are a large number of papers and software tools
dedicated to simulating either fluid flow in 3D computerized tomography (CT)
images or reactive flow using pore-network models, little attention to date has
been focused on the pore-scale simulation of sorptive transport in 3D CT
images, which is the specific focus of this paper. Here we first present an
algorithm for the simulation of such reactive flows directly on images, which
is implemented in a sophisticated software package. We then use this software
to present numerical results in two resolved geometries, illustrating the
importance of pore-scale simulation and the flexibility of our software
package.Comment: 15 pages, 6 figure
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